![]() |
Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > madjusmdet | Structured version Visualization version GIF version |
Description: Express the cofactor of the matrix, i.e. the entries of its adjunct matrix, using determinant of submatrices. (Contributed by Thierry Arnoux, 23-Aug-2020.) |
Ref | Expression |
---|---|
madjusmdet.b | ⊢ 𝐵 = (Base‘𝐴) |
madjusmdet.a | ⊢ 𝐴 = ((1...𝑁) Mat 𝑅) |
madjusmdet.d | ⊢ 𝐷 = ((1...𝑁) maDet 𝑅) |
madjusmdet.k | ⊢ 𝐾 = ((1...𝑁) maAdju 𝑅) |
madjusmdet.t | ⊢ · = (.r‘𝑅) |
madjusmdet.z | ⊢ 𝑍 = (ℤRHom‘𝑅) |
madjusmdet.e | ⊢ 𝐸 = ((1...(𝑁 − 1)) maDet 𝑅) |
madjusmdet.n | ⊢ (𝜑 → 𝑁 ∈ ℕ) |
madjusmdet.r | ⊢ (𝜑 → 𝑅 ∈ CRing) |
madjusmdet.i | ⊢ (𝜑 → 𝐼 ∈ (1...𝑁)) |
madjusmdet.j | ⊢ (𝜑 → 𝐽 ∈ (1...𝑁)) |
madjusmdet.m | ⊢ (𝜑 → 𝑀 ∈ 𝐵) |
Ref | Expression |
---|---|
madjusmdet | ⊢ (𝜑 → (𝐽(𝐾‘𝑀)𝐼) = ((𝑍‘(-1↑(𝐼 + 𝐽))) · (𝐸‘(𝐼(subMat1‘𝑀)𝐽)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | madjusmdet.b | . 2 ⊢ 𝐵 = (Base‘𝐴) | |
2 | madjusmdet.a | . 2 ⊢ 𝐴 = ((1...𝑁) Mat 𝑅) | |
3 | madjusmdet.d | . 2 ⊢ 𝐷 = ((1...𝑁) maDet 𝑅) | |
4 | madjusmdet.k | . 2 ⊢ 𝐾 = ((1...𝑁) maAdju 𝑅) | |
5 | madjusmdet.t | . 2 ⊢ · = (.r‘𝑅) | |
6 | madjusmdet.z | . 2 ⊢ 𝑍 = (ℤRHom‘𝑅) | |
7 | madjusmdet.e | . 2 ⊢ 𝐸 = ((1...(𝑁 − 1)) maDet 𝑅) | |
8 | madjusmdet.n | . 2 ⊢ (𝜑 → 𝑁 ∈ ℕ) | |
9 | madjusmdet.r | . 2 ⊢ (𝜑 → 𝑅 ∈ CRing) | |
10 | madjusmdet.i | . 2 ⊢ (𝜑 → 𝐼 ∈ (1...𝑁)) | |
11 | madjusmdet.j | . 2 ⊢ (𝜑 → 𝐽 ∈ (1...𝑁)) | |
12 | madjusmdet.m | . 2 ⊢ (𝜑 → 𝑀 ∈ 𝐵) | |
13 | eqeq1 2732 | . . . 4 ⊢ (𝑘 = 𝑖 → (𝑘 = 1 ↔ 𝑖 = 1)) | |
14 | breq1 5151 | . . . . 5 ⊢ (𝑘 = 𝑖 → (𝑘 ≤ 𝐼 ↔ 𝑖 ≤ 𝐼)) | |
15 | oveq1 7427 | . . . . 5 ⊢ (𝑘 = 𝑖 → (𝑘 − 1) = (𝑖 − 1)) | |
16 | id 22 | . . . . 5 ⊢ (𝑘 = 𝑖 → 𝑘 = 𝑖) | |
17 | 14, 15, 16 | ifbieq12d 4557 | . . . 4 ⊢ (𝑘 = 𝑖 → if(𝑘 ≤ 𝐼, (𝑘 − 1), 𝑘) = if(𝑖 ≤ 𝐼, (𝑖 − 1), 𝑖)) |
18 | 13, 17 | ifbieq2d 4555 | . . 3 ⊢ (𝑘 = 𝑖 → if(𝑘 = 1, 𝐼, if(𝑘 ≤ 𝐼, (𝑘 − 1), 𝑘)) = if(𝑖 = 1, 𝐼, if(𝑖 ≤ 𝐼, (𝑖 − 1), 𝑖))) |
19 | 18 | cbvmptv 5261 | . 2 ⊢ (𝑘 ∈ (1...𝑁) ↦ if(𝑘 = 1, 𝐼, if(𝑘 ≤ 𝐼, (𝑘 − 1), 𝑘))) = (𝑖 ∈ (1...𝑁) ↦ if(𝑖 = 1, 𝐼, if(𝑖 ≤ 𝐼, (𝑖 − 1), 𝑖))) |
20 | breq1 5151 | . . . . 5 ⊢ (𝑘 = 𝑖 → (𝑘 ≤ 𝑁 ↔ 𝑖 ≤ 𝑁)) | |
21 | 20, 15, 16 | ifbieq12d 4557 | . . . 4 ⊢ (𝑘 = 𝑖 → if(𝑘 ≤ 𝑁, (𝑘 − 1), 𝑘) = if(𝑖 ≤ 𝑁, (𝑖 − 1), 𝑖)) |
22 | 13, 21 | ifbieq2d 4555 | . . 3 ⊢ (𝑘 = 𝑖 → if(𝑘 = 1, 𝑁, if(𝑘 ≤ 𝑁, (𝑘 − 1), 𝑘)) = if(𝑖 = 1, 𝑁, if(𝑖 ≤ 𝑁, (𝑖 − 1), 𝑖))) |
23 | 22 | cbvmptv 5261 | . 2 ⊢ (𝑘 ∈ (1...𝑁) ↦ if(𝑘 = 1, 𝑁, if(𝑘 ≤ 𝑁, (𝑘 − 1), 𝑘))) = (𝑖 ∈ (1...𝑁) ↦ if(𝑖 = 1, 𝑁, if(𝑖 ≤ 𝑁, (𝑖 − 1), 𝑖))) |
24 | eqeq1 2732 | . . . 4 ⊢ (𝑙 = 𝑗 → (𝑙 = 1 ↔ 𝑗 = 1)) | |
25 | breq1 5151 | . . . . 5 ⊢ (𝑙 = 𝑗 → (𝑙 ≤ 𝐽 ↔ 𝑗 ≤ 𝐽)) | |
26 | oveq1 7427 | . . . . 5 ⊢ (𝑙 = 𝑗 → (𝑙 − 1) = (𝑗 − 1)) | |
27 | id 22 | . . . . 5 ⊢ (𝑙 = 𝑗 → 𝑙 = 𝑗) | |
28 | 25, 26, 27 | ifbieq12d 4557 | . . . 4 ⊢ (𝑙 = 𝑗 → if(𝑙 ≤ 𝐽, (𝑙 − 1), 𝑙) = if(𝑗 ≤ 𝐽, (𝑗 − 1), 𝑗)) |
29 | 24, 28 | ifbieq2d 4555 | . . 3 ⊢ (𝑙 = 𝑗 → if(𝑙 = 1, 𝐽, if(𝑙 ≤ 𝐽, (𝑙 − 1), 𝑙)) = if(𝑗 = 1, 𝐽, if(𝑗 ≤ 𝐽, (𝑗 − 1), 𝑗))) |
30 | 29 | cbvmptv 5261 | . 2 ⊢ (𝑙 ∈ (1...𝑁) ↦ if(𝑙 = 1, 𝐽, if(𝑙 ≤ 𝐽, (𝑙 − 1), 𝑙))) = (𝑗 ∈ (1...𝑁) ↦ if(𝑗 = 1, 𝐽, if(𝑗 ≤ 𝐽, (𝑗 − 1), 𝑗))) |
31 | breq1 5151 | . . . . 5 ⊢ (𝑙 = 𝑗 → (𝑙 ≤ 𝑁 ↔ 𝑗 ≤ 𝑁)) | |
32 | 31, 26, 27 | ifbieq12d 4557 | . . . 4 ⊢ (𝑙 = 𝑗 → if(𝑙 ≤ 𝑁, (𝑙 − 1), 𝑙) = if(𝑗 ≤ 𝑁, (𝑗 − 1), 𝑗)) |
33 | 24, 32 | ifbieq2d 4555 | . . 3 ⊢ (𝑙 = 𝑗 → if(𝑙 = 1, 𝑁, if(𝑙 ≤ 𝑁, (𝑙 − 1), 𝑙)) = if(𝑗 = 1, 𝑁, if(𝑗 ≤ 𝑁, (𝑗 − 1), 𝑗))) |
34 | 33 | cbvmptv 5261 | . 2 ⊢ (𝑙 ∈ (1...𝑁) ↦ if(𝑙 = 1, 𝑁, if(𝑙 ≤ 𝑁, (𝑙 − 1), 𝑙))) = (𝑗 ∈ (1...𝑁) ↦ if(𝑗 = 1, 𝑁, if(𝑗 ≤ 𝑁, (𝑗 − 1), 𝑗))) |
35 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 19, 23, 30, 34 | madjusmdetlem4 33431 | 1 ⊢ (𝜑 → (𝐽(𝐾‘𝑀)𝐼) = ((𝑍‘(-1↑(𝐼 + 𝐽))) · (𝐸‘(𝐼(subMat1‘𝑀)𝐽)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 ifcif 4529 class class class wbr 5148 ↦ cmpt 5231 ‘cfv 6548 (class class class)co 7420 1c1 11139 + caddc 11141 ≤ cle 11279 − cmin 11474 -cneg 11475 ℕcn 12242 ...cfz 13516 ↑cexp 14058 Basecbs 17179 .rcmulr 17233 CRingccrg 20173 ℤRHomczrh 21424 Mat cmat 22306 maDet cmdat 22485 maAdju cmadu 22533 subMat1csmat 33394 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 ax-addf 11217 ax-mulf 11218 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-xor 1506 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-ot 4638 df-uni 4909 df-int 4950 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-isom 6557 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-of 7685 df-om 7871 df-1st 7993 df-2nd 7994 df-supp 8166 df-tpos 8231 df-frecs 8286 df-wrecs 8317 df-recs 8391 df-rdg 8430 df-1o 8486 df-2o 8487 df-er 8724 df-map 8846 df-pm 8847 df-ixp 8916 df-en 8964 df-dom 8965 df-sdom 8966 df-fin 8967 df-fsupp 9386 df-sup 9465 df-oi 9533 df-card 9962 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-div 11902 df-nn 12243 df-2 12305 df-3 12306 df-4 12307 df-5 12308 df-6 12309 df-7 12310 df-8 12311 df-9 12312 df-n0 12503 df-xnn0 12575 df-z 12589 df-dec 12708 df-uz 12853 df-rp 13007 df-fz 13517 df-fzo 13660 df-seq 13999 df-exp 14059 df-hash 14322 df-word 14497 df-lsw 14545 df-concat 14553 df-s1 14578 df-substr 14623 df-pfx 14653 df-splice 14732 df-reverse 14741 df-s2 14831 df-struct 17115 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17180 df-ress 17209 df-plusg 17245 df-mulr 17246 df-starv 17247 df-sca 17248 df-vsca 17249 df-ip 17250 df-tset 17251 df-ple 17252 df-ds 17254 df-unif 17255 df-hom 17256 df-cco 17257 df-0g 17422 df-gsum 17423 df-prds 17428 df-pws 17430 df-mre 17565 df-mrc 17566 df-acs 17568 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-mhm 18739 df-submnd 18740 df-efmnd 18820 df-grp 18892 df-minusg 18893 df-mulg 19023 df-subg 19077 df-ghm 19167 df-gim 19212 df-cntz 19267 df-oppg 19296 df-symg 19321 df-pmtr 19396 df-psgn 19445 df-cmn 19736 df-abl 19737 df-mgp 20074 df-rng 20092 df-ur 20121 df-ring 20174 df-cring 20175 df-oppr 20272 df-dvdsr 20295 df-unit 20296 df-invr 20326 df-dvr 20339 df-rhm 20410 df-subrng 20482 df-subrg 20507 df-drng 20625 df-sra 21057 df-rgmod 21058 df-cnfld 21279 df-zring 21372 df-zrh 21428 df-dsmm 21665 df-frlm 21680 df-mat 22307 df-marrep 22459 df-subma 22478 df-mdet 22486 df-madu 22535 df-minmar1 22536 df-smat 33395 |
This theorem is referenced by: mdetlap 33433 |
Copyright terms: Public domain | W3C validator |